Gov.uk says GPs cannot provide letters showing your Covid-19 vaccination status.
These are the current options:
Do you have a smartphone or iPad/Tablet ?
You can download the NHS App.
There is also a YouTube Quick Guide to getting started.
No smartphone or smart device ?
You can use the NHS App via a computer or laptop.
No devices? No Internet ?
From 17th May
Call NHS 119 and they will be able to post a letter to you. You will need to provide details of the country you are travelling to for this.
Please see Gov.uk: Demonstrating your Covid-19 Status when travelling abroad for more information.